Abstract. Defining the standard Boolean operations on fuzzy Booleans with the compositional rule of inference (CRI) or Zadeh's extension principle gives.
The Compositional Rule of Inference and Zadeh's Extension Principle for Non-normal Fuzzy Sets Pim van den Broek1 and Joost Noppen1 1Dept. of Computer Science, University of Twente P.O. Box 217, 7500 AE Enschede, The Netherlands {pimvdb,noppen}@ewi.utwente.nl
Abstract. Defining the standard Boolean operations on fuzzy Booleans with the compositional rule of inference (CRI) or Zadeh's extension principle gives counter-intuitive results. We introduce and motivate a slight adaptation of the CRI, which only effects the results for non-normal fuzzy sets. It is shown that the adapted CRI gives the expected results for the standard Boolean operations on fuzzy Booleans. As a second application, we show that the adapted CRI enables a don't-care value in approximate reasoning. From the close connection between the CRI and Zadeh's extension principle, we derive an adaptation of the extension principle, which, like the modified CRI, also gives the expected Boolean operations on fuzzy Booleans. Keywords: compositional rule of inference, extension principle, fuzzy Booleans
1 Introduction Fuzzy Booleans are introduced in [1], in analogy with the concept of fuzzy numbers [2], as fuzzy sets over the domain of truth-values {true, false}. A fuzzy Boolean is denoted as (a,b), where a and b are numbers from the interval [0,1], a shorthand for the conventional notation "a/true + b/false". The truth-values 'true' and 'false' are represented by (1,0) and (0,1) respectively. The advantage of fuzzy booleans is that they allow fuzzy reasoning with concepts 'contradiction' and 'undefined'. For instance, when interpreted as possibilities, (1,1) is 'undefined', and (0,0) is 'contradiction'; when interpreted as necessities, this is just the other way around. Let AND and OR be the fuzzy equivalents of the crisp Boolean operators and and or, respectively. They may be defined by means of approximate reasoning. For instance, AND is defined by the following four fuzzy rules: IF X1=(1,0) AND X2=(1,0) THEN Y=(1,0) IF X1=(1,0) AND X2=(0,1) THEN Y=(0,1) IF X1=(0,1) AND X2=(1,0) THEN Y=(0,1) IF X1=(0,1) AND X2=(0,1) THEN Y=(0,1) . With approximate reasoning, using the CRI, we obtain
(1a) (1b) (1c) (1d)
AND ((a,b),(c,d)) = (min(a,c), max {min (a,d), min(b,c), min(b,d)}) .
(2)
Using the abbreviations T = (1,0) (True), F = (0,1) (False), C = (1,1) (Contradiction) and U = (0,0) (Unknown), we obtain for AND: AND | T F C U --------------------------------------T | T F C U F | F F F U C | C F C U U | U U U U. This is not in accordance with our intuitive understanding of the AND-operation. For instance, AND (F,U) = U, where we would expect AND (F,U) = F. Indeed, since U is the empty set, the CRI will give U whenever one of the arguments of AND is U. The same thing happens when we define AND with Zadeh's extension principle: whenever one of the arguments of AND is U, the result is U. The aim of this paper is to introduce and motivate a slight adaptation of the CRI, which only effects the results for non-normal fuzzy sets, and show that the adapted CRI gives the expected results for the standard Boolean operations on fuzzy Booleans. As a second application, we will show that the adapted CRI enables a don't-care value in approximate reasoning. From the close connection between the CRI and Zadeh's extension principle, we will derive an adaptation of the extension principle, which, like the modified CRI, also gives the expected Boolean operations on fuzzy Booleans. This paper is organised as follows. In the next section, we will introduce the adaptation of the CRI. In section 3 we describe approximate reasoning with the adapted CRI. In section 4 we show that the adapted CRI gives results for the standard Boolean operations on fuzzy Booleans which are in accordance with our intuitive understanding of the AND-operation. In section 5 we show that the adapted CRI enables a don't-care value in approximate reasoning. In section 6 we derive a adaptation of the extension principle, and show that with the adapted extension principle we also obtain the expected Boolean operations on fuzzy Booleans. Section 7 concludes the paper.
2 Adaptation of the compositional rule of inference Given a fuzzy set A on a domain U and a fuzzy relation R on the domain U⊗V, the CRI gives the fuzzy set B on V which is given by B(v) = supu min (A(u), R(u,v)) .
(3)
If A is non-normal, B is non-normal as well. If A is empty, B is empty as well. If A is the crisp singleton set containing only u0, then B(v) = R(u0,v). So, for each crisp singleton set A
B(v) >= infu R(u,v) .
(4)
Our adaptation of the CRI is such that eq. (4) holds for every fuzzy set A. Instead of eq. (3), we thus propose B(v) = supu max(infu R(u,v), min (A(u), R(u,v)) .
(5)
This adaptation can only give a different result when A is non-normal. Indeed, if A(u0) = 1 for some u0 in U then supu min (A(u), R(u,v)) >= min (A(u0), R(u0,v)) = R(u0,v) >= infu R(u,v), and so eq. (5) reduces to eq. (3). Also, when R(u,v) = 0 for some u in U there is no difference between eq. (5) and eq. (3). Next we will define how to adapt a composition of two applications of the CRI as in B(v) = sup(u1,u2) min(A1(u1), min (A2(u2), R((u1,u2),v))) .
(6)
Here it is not appropriate to adapt this in a single step by first writing this as B(v) = supu min(A1⊗A2 (u), R(u,v)))
(7)
where A1⊗A2 is the Cartesean product of A1 and A2 and u = (u1,u2). Instead, the adaptation should be applied twice, which leads to B(v) = sup(u1,u2) max(infu1 max(infu2 R((u1,u2),v), min(A2(u2), R((u1,u2),v))), min(A1(u1), max(infu2 R((u1,u2),v), min(A2 (u2), R((u1,u2),v))))) . (8)
3 Approximate reasoning with the adapted CRI Consider the fuzzy rule IF X = A' THEN Y = B' .
(9)
Given the fact X = A, approximate reasoning with the standard CRI gives Y = B, where B is given by eq. (3), and R(u,v) is given by R(u,v) = Q (A'(u), B'(v)) .
(10)
In case of approximate reasoning with the interpolation method [4], the operator Q is a t-norm; the most commonly used t-norm is the minimum operator. In case of approximate reasoning with the implication method [3], the operator Q is an implication operator. In case of multiple fuzzy rules, one calculates a relation as in eq. (10) for each fuzzy rule, and then aggregates the results. Aggregation is done with the maximum operator in the interpolation method, and with the minimum operator in the
implication method. The resulting aggregated relation is then used to calculate the inference results with eq. (3). Using the adapted CRI means that eq. (5) should be used instead of eq. (3). Consider next the fuzzy rule with two antecedents IF X1 = A'1 AND X2 =A'2 THEN Y = B' .
(11)
With the standard CRI we have, given the input X1 = A1 AND X2 =A2, the output Y = B, where B is given by eq. (6), and R((u1,u2),v) = Q (min (A'1(u1), A'2(u2)), B'(v))
(12)
In case of multiple fuzzy rules, one calculates a relation as in eq. (12) for each fuzzy rule, and then aggregates the results. The resulting aggregated relation is then used to calculate the inference results with eq. (6). Using the adapted CRI means that eq. (8) should be used instead of eq. (6). Generalisation to three or more antecedents is straightforward. Note that we described here the FATI (first aggregate, then inference) approach. In Mamdani's original approach [4], the interpolation method with the minimum operator as t-norm, the FITA (first inference, then aggregate) approach is used. Indeed, it happens that the inference results of FATI and FITA are the same in this case. This is no longer true when the adapted CRI is used. So, with the adapted CRI, one should always adopt the FATI approach.
4 Application to fuzzy Booleans In this section we will use the results of the previous section to compute the inference results for the four fuzzy rules of eq. (1) with the adapted CRI. First we compute the four relations R1, R2, R3 and R4 for the fuzzy rules of eqs. (1a,1b,1c,1d) respectively, and their aggregation R in the interpolation method, using eq. (12): | ((t,t),t) ((t,f),t) ((f,t),t) ((f,f),t) (t,t),f) ((t,f),f) ((f,t),f) ((f,f),f) ---------------------------------------------------------------------------------------R1 | 1 0 0 0 0 0 0 0 R2 | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 R3 | 0 R4 | 0 0 0 0 0 0 0 1 R | 1 0 0 0 0 1 1 1 Here we used the abbreviations t and f for true and false, respectively. Next we compute the four relations R1, R2, R3 and R4 and their aggregation R in the implication method:
| ((t,t),t) ((t,f),t) ((f,t),t) ((f,f),t) (t,t),f) ((t,f),f) ((f,t),f) ((f,f),f) ---------------------------------------------------------------------------------------R1 | 1 1 1 1 0 1 1 1 R2 | 1 0 1 1 1 1 1 1 R3 | 1 1 0 1 1 1 1 1 R4 | 1 1 1 0 1 1 1 1 R | 1 0 0 0 0 1 1 1 Note that the aggregated relation R is the same for both methods, and is independent of Q. This is in accordance with in general result in [1], where it is proved that this is always the case when the set of fuzzy rules is a complete set of fuzzy rules with crisp antecedents. Substituting this relation in eq. (8) now gives AND ((a,b),(c,d)) = (min(a,c), max (b,d))
(13)
whereas with the standard CRI (eq. (6)) we would have obtained eq.(2). We can verify that eq. (13) is in accordance with our intuitive understanding of the ANDoperation. Indeed, eq. (13) just says that the "trueness" of AND P Q is the trueness of both P and Q, and the "falseness" of AND P Q is the falseness of either P or Q. This should be compared with eq. (2), where the falseness of P does not imply the falseness of AND P Q; the falseness of AND P Q follows only if in addition to the falseness of P we also have either the trueness or the falseness of Q. The table in the introduction is replaced by AND | T F C U --------------------------------------T | T F C U F | F F F F C | C F C F U | U F F U. In the same way, we obtain the expression OR ((a,b),(c,d)) = (max (a,c),min (b,d)) and the table OR | T F C U --------------------------------------T | T T T T F | T F C U C | T C C T U | T U T U
(14)
which are in accordance with our intuitive understanding of the OR-operation. Finally, the NOT-operation, given by NOT (a,b) = (b,a), is not affected by our adaptation of the CRI.
5 Don't-care value in approximate reasoning As a second application of the adapted CRI, we will show in this section that with the adapted CRI there exists a don't-care value in approximate reasoning. Consider first the fuzzy rule of eq. (9). Given the fact X = A, approximate reasoning with the CRI gives Y = B, where B is given by B(v) = supu min (A(u), Q (A'(u),B'(v))) .
(15)
We will consider first the interpolation method, i.e. Q is a t-norm. The fuzzy set A' is a don't-care value if B(v) = B'(v) for all v in V and all fuzzy sets A. Since B(v)
